354 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. XII 



then A(2xy) 2 + 1 = D, so that we obtain an infinitude of solutions, but 

 not all, from one solution. A. Marre 60 stated that the last result was 

 copied from a letter written by Claude Jaquemet, who gave the second 

 solution X = 2xy, Y = 2 Ax 2 + 1. 



Wallis 61 attempted to prove that t 2 Du 2 = 1 always has positive 

 integral solutions, but made use of a lemma which is false [Lagrange 74 - 85 

 and Gauss 93 ]]: Let m be the integer just > VZ>, whence m VD < 1, and 

 set p = m VZ>, r = 1/(2VZ)); then it is possible to find two integers z 

 and a such that 



z_ 



a 2a 



But the difference of the fractions in this inequality approaches zero as 

 z and a increase, so that their ratio approaches p. 



The name Pell equation for x 2 Dy 2 = 1 originated in the erroneous 

 notion of L. Euler 62 that John Pell was the author of the unique method of 

 solution explained in Wallis' Opera, whereas Wallis gave only Brouncker's 

 method. Nor, as stated by Hankel, 24 had Pell treated the equation in a 

 widely read work, i. e., in his notes to Brancker's 63 English translation of 

 J. H. Rahn's algebra. After examining three copies of this translation, 

 G. Enestrom 64 stated that there is nothing relating to this equation. How- 

 ever, x I2y 2 z 2 is treated in Rahn's 63 Algebra, p. 143. 



Euler 62 noted that if az~ -\- bz + c is a square I 2 for z = p, it is a square 

 for 



so that the problem is to make 1 + aX 2 a square. 



Euler 65 again noted that, if / = ax 2 + bx -f- c is a square m 2 for x = n, 

 it is the square of m' = apn + pb/2 + qmiorx = qn + pm + (bq 6)/(2a), 

 provided that q 2 = ap 2 + 1. In the latter expression for x we replace n 

 by this x and replace m by m' and get 



x' = 2q 2 n + 2pqm + - (q 2 - 1) - n, 



Cv 



which makes / = D. If A, B are consecutive terms of the series n, x, x', 

 , the next term is 2qB A + b(q 1)1 a. In the case / = ax 2 + 1, 

 whence 6 = 0, c = 1, the series becomes 0, p, 2pq, 4pg 2 p, - -, A, B, 



60 Bull. Bibl. Storia Sc. Mat. Fis., 12, 1879, 893. Attributed incorrectly to Marquis de 



1'Hopital in Comptes Rendus Paris, 88, 1879, 76-7, 223. 



61 Algebra, Oxford, 1685, Ch. 99; Opera, 2, 1693, 427-8. Reproduced by Konen, 3 43-6. 



62 Letter to Goldbach, Aug. 10, 1730, Correspondance Math, et Physique (ed., P. H. Fuss), 



St. Petersburg, 1, 1843, 37. Also, Euler.' 5 ' 72 Cf. Euler 56 of Ch. XIII. Cf. P. Tan- 

 nery, Bull, des Sc. Math., (2), 27, I, 47-9. 



63 An introduction to algebra, translated out of the High Dutch into English by T. Brancker. 



Much altered and augmented by D. P. London, 1668. On Rahn's algebra of 1659, see 



Bibliotheca Math., (3), 3, 1902, 125. 



M Bibliotheca Math., (3), 3, 1902, 204; cf. G. Wertheim, 2, 1901, 360-1. 

 66 Comm. Acad. Petrop., 6, 1732-3, 175; Comm. Arith. Coll., 1, 1849, 4; Op. Om., (1), II, 6 



